We derive and study necessary and sufficient conditions for an S-bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection me...

In this paper, inspired by P.M.H. Wilson’s paper on sectional curvatures of Kähler moduli [31], we concentrate on the case where f is a homogeneous polynomial (also called a “form”) of degree d at least 2. Following Okonek and van de Ven [23], Wilson considers the “index cone,” the open subset where the Hessian matrix of f is Lorentzian (that is, of signature (1, ∗)) and f is positive. He restr...

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature KM (x) ≥ − C dp(x) is diffeomorphic to a Euclidean n-space R under some conditions on the density of rays starting from the base point p or on the volum...

In this paper, we have proved that locally there exist infinitely many three dimensional slant submanifolds with prescribed scalar curvature into cosymplectic space form M 5 (c) with c ∈ {−4, 4}while there does not exist flat minimal proper slant surface in M 5 (c) with c 6= 0. In section 5, we have established an inequality between mean curvature and sectional curvature of the subamnifold and ...

A type of almost contact hypersurfaces with Norden metric of a Kähler manifold with Norden metric is considered. The curvature tensor and the special sectional curvatures are characterized. The canonical connection on such manifolds is studied and the form of the corresponding Kähler curvature tensor is obtained. Some curvature properties of the manifolds belonging to the widest integrable main...

In [4], R. Hamilton has proved that if a compact manifold M of dimension three admits a C Riemannian metric g0 with positive Ricci curvature, then it also admits a metric g with constant positive sectional curvature, and is thus a quotient of the sphere S. In fact, he shows that the original metric can be deformed into the constant-curvature metric by requiring that, for t ≥ 0, x ∈ M and g = g(...

The classic 2π-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the “2π-metric” and the hyperbolic metric. We make partial progress in the program, provi...

We classify the complete metrics of nonnegative sectional curvature on M × R, where M is any compact 2-manifold.